Cyclic characters of symmetric groups (Q1592960)

Let \(S_n\) denote the symmetric group of degree \(n\) and \(C\) denote a subgroup of \(S_n\) generated by a cycle \(\sigma\) of order \(n\). Then \(\psi_i\colon C\to\mathbb{C}^*\) defined by \(\psi_i(\sigma)=\xi^i\), where \(\xi\) is a primitive \(n\)-th root of unity in the complex field \(\mathbb{C}\), is an irreducible ordinary character of \(C\). It is well-known that there is a one-to-one correspondence between the complex irreducible characters of \(S_n\) and the partitions of \(n\). Now, if \(p\) is a partition of \(n\), then the irreducible character of \(S_n\) corresponding to \(p\) is denoted by \(\xi^p\). In the paper under review the authors prove that the multiplicity \((\psi^{S_n}_i,\xi^p)\) is equal to the number of standard Young tableaux of shape \(p\) and its major index is congruent to \(i\) modulo \(n\). This fact was already proved by \textit{W. Kraśkiewicz} and \textit{J. Weyman} [Bayreuther Math. Schr. 63, 265-284 (2001; Zbl 1037.20012)]. In the present paper the authors give a new proof.